A154372 -id:A154372 - cp's OEIS frontend

A080108 a(n) = Sum_{k=1..n} k^(n-k)*binomial(n-1,k-1).

1, 2, 6, 23, 104, 537, 3100, 19693, 136064, 1013345, 8076644, 68486013, 614797936, 5818490641, 57846681092, 602259154853, 6548439927680, 74180742421185, 873588590481988, 10674437936521069, 135097459659312176

Offset: 1

Vladeta Jovovic, Mar 15 2003

nonn

Row sums of triangle A154372. Example: a(3)=1+12+9+1=23. From A152818. - Paul Curtz, Jan 08 2009
Number of pointed set partitions of a pointed set k[1...k...n] with a prescribed point k. - Gus Wiseman, Sep 27 2015
With offset 0, a(n) is the number of partial functions (A000169) from [n]->[n] such that every element  in the domain of definition is mapped to a fixed point.  This implies a(n) is the number of idempotent partial functions Cf. A121337. - Geoffrey Critzer, Aug 07 2016

			G.f. = x + 2*x^2 + 6*x^3 + 23*x^4 + 104*x^5 + 537*x^6 + 3100*x^7 + 19693*x^8 + ...
The a(4) = 23 pointed set partitions of 1[1 2 3 4] are 1[1[1 2 3 4]], 1[1[1] 2[2 3 4]], 1[1[1] 3[2 3 4]], 1[1[1] 4[2 3 4]], 1[1[1 2] 3[3 4]], 1[1[1 2] 4[3 4]], 1[1[1 3] 2[2 4]], 1[1[1 3] 4[2 4]], 1[1[1 4] 2[2 3]], 1[1[1 4] 3[2 3]], 1[1[1 2 3] 4[4]], 1[1[1 2 4] 3[3]], 1[1[1 3 4] 2[2]], 1[1[1] 2[2] 3[3 4]], 1[1[1] 2[2] 4[3 4]], 1[1[1] 2[2 3] 4[4]], 1[1[1] 2[2 4] 3[3]], 1[1[1] 3[3] 4[2 4]], 1[1[1] 3[2 3] 4[4]], 1[1[1 2] 3[3] 4[4]], 1[1[1 3] 2[2] 4[4]], 1[1[1 4] 2[2] 3[3]], 1[1[1] 2[2] 3[3] 4[4]].

Vincenzo Librandi, Table of n, a(n) for n = 1..200

First column of array A098697.

Cf. A152818, A185298, A262671.

[(1/n)*(&+[Binomial(n,k)*k^(n-k+1): k in [0..n]]): n in [1..30]]; // G. C. Greubel, Jan 22 2023

Table[Sum[k^(n-k) Binomial[n-1,k-1],{k,n}],{n,30}] (* Harvey P. Dale, Aug 19 2012 *)
Table[SeriesCoefficient[Sum[x^k/(1-k*x)^k,{k,0,n}],{x,0,n}], {n,1,20}] (* Vaclav Kotesovec, Aug 06 2014 *)
CoefficientList[Series[E^(x*(1+E^x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Aug 06 2014 *)

a(n)=sum(k=1,n, k^(n-k)*binomial(n-1,k-1)) \\ Anders Hellström, Sep 27 2015

def A080108(n): return (1/n)*sum(binomial(n,k)*k^(n-k+1) for k in range(n+1))
[A080108(n) for n in range(1,31)] # G. C. Greubel, Jan 22 2023

G.f.: Sum_{k>0} x^k/(1-k*x)^k.
E.g.f. (for offset 0): exp(x*(1+exp(x))). - Vladeta Jovovic, Aug 25 2003
a(n) = A185298(n)/n.

A072597 Expansion of 1/(exp(-x) - x) as exponential generating function.

Original entry on oeis.org

1, 2, 7, 37, 261, 2301, 24343, 300455, 4238153, 67255273, 1185860331, 23000296155, 486655768525, 11155073325917, 275364320099807, 7282929854486431, 205462851526617489, 6158705454187353297, 195465061563672788947, 6548320737474275229347, 230922973019493881984021

Offset: 0

Michael Somos, Jun 23 2002

Polynomials from A140749/A141412 are linked to Stirling1 (see A048594, A129841, A140749). See also P. Flajolet, X. Gourdon, B. Salvy in, available on Internet, RR-1857.pdf (preprint of unavailable Gazette des Mathematiciens 55, 1993, pp. 67-78; for graph 2 see also X. Gourdon RR-1852.pdf, pp. 64-65). What is the corresponding graph for A152650/A152656 = simplified A009998/A119502 linked, via A152818, to a(n), then Stirling2? - Paul Curtz, Dec 16 2008
Denominators in rational approximations of Lambert W(1). See Ramanujan, Notebooks, volume 2, page 22: "2. If e^{-x} = x, shew that the convergents to x are 1/2, 4/7, 21/37, 148/261, &c." Numerators in A006153. - Michael Somos, Jan 21 2019
Call an element g in a semigroup a group element if g^j = g for some j > 1. Then a(n) is the number of group elements in the semigroup of partial transformations of an n-set. Hence a(n) = Sum_{k=0..n} A154372(n,k)*k!. - Geoffrey Critzer, Nov 27 2021

			G.f. = 1 + 2*x + 7*x^2 + 37*x^3 + 261*x^4 + 2301*x^5 + 24343*x^6 + ...

O. Ganyushkin and V Mazorchuk, Classical Finite Transformation Semigroups, Springer, 2009, page 70.
S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 2, see page 22.

Seiichi Manyama, Table of n, a(n) for n = 0..411
W. S. Gray and M. Thitsa, System Interconnections and Combinatorial Integer Sequences, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-11 March 2013.
G. Jiraskova and J. Shallit, The state complexity of star-complement-star, arXiv preprint arXiv:1203.5353 [cs.FL], 2012. - From _N. J. A. Sloane_, Sep 21 2012

Cf. A000110, A006153, A030178, A089148.
Cf. A048993, A052820.
Cf. A006153, A154372.

CoefficientList[Series[1/(Exp[-x]-x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 26 2013 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ 1 / (Exp[-x] - x), {x, 0, n}]]; (* Michael Somos, Jan 21 2019 *)
a[ n_] := If[ n < 0, 0, n! Sum[ (n - k + 1)^k / k!, {k, 0, n}]]; (* Michael Somos, Jan 21 2019 *)

{a(n) = if( n<0, 0, n! * polcoeff( 1 / (exp(-x + x * O(x^n)) - x), n))};

{a(n) = if( n<0, 0, n! * sum(k=0, n, (n-k+1)^k / k!))}; /* Michael Somos, Jan 21 2019 */

E.g.f.: 1 / (exp(-x) - x).
a(n) = n!*Sum_{k=0..n} (n-k+1)^k/k!. - Vladeta Jovovic, Aug 31 2003
a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling2(n, k)*A052820(k). - Vladeta Jovovic, Apr 12 2004
Recurrence: a(n+1) = 1 + Sum_{j=1..n} binomial(n, j)*a(j)*j. - Jon Perry, Apr 25 2005
E.g.f.: 1/(Q(0) - x) where Q(k) = 1 - x/(2*k+1 - x*(2*k+1)/(x - (2*k+2)/Q(k+1) )); (continued fraction ). - Sergei N. Gladkovskii, Apr 04 2013
a(n) ~ n!/((1+c)*c^(n+1)), where c = A030178 = LambertW(1) = 0.5671432904... - Vaclav Kotesovec, Jun 26 2013
O.g.f.: Sum_{k>=0} k!*x^k/(1 - (k + 1)*x)^(k+1). - Ilya Gutkovskiy, Oct 09 2018
a(n) = A006153(n+1)/(n+1). - Seiichi Manyama, Nov 05 2024

A152818 Array read by antidiagonals: A(n,k) = (k+1)^n*(n+k)!/n!.

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 1, 12, 18, 6, 1, 32, 108, 96, 24, 1, 80, 540, 960, 600, 120, 1, 192, 2430, 7680, 9000, 4320, 720, 1, 448, 10206, 53760, 105000, 90720, 35280, 5040, 1, 1024, 40824, 344064, 1050000, 1451520, 987840, 322560, 40320

Offset: 0

Paul Curtz, Dec 13 2008

A009998/A119502 gives triangle of unreduced coefficients of polynomials defined by A152650/A152656. a(n) gives numerators with denominators n! for each row.

Row 0 is A000142. Row 1 is formed from positive members of A001563. Row 2 is A055533. Column 0 is A000012. Column 1 is formed from positive members of A001787. Column 2 is A006043. Column 3 is A006044. - Omar E. Pol, Jan 06 2009

			From _Omar E. Pol_, Jan 06 2009: (Start)
Array begins:
  1,    1,      2,        6,         24,          120, ...
  1,    4,     18,       96,        600,         4320, ...
  1,   12,    108,      960,       9000,        90720, ...
  1,   32,    540,     7680,     105000,      1451520, ...
  1,   80,   2430,    53760,    1050000,     19595520, ...
  1,  192,  10206,   344064,    9450000,    235146240, ...
  1,  448,  40824,  2064384,   78750000,   2586608640, ...
  1, 1024, 157464, 11796480,  618750000,  26605117440, ...
  1, 2304, 590490, 64880640, 4640625000, 259399895040, ... (End)
Antidiagonal triangle:
  1;
  1,   1;
  1,   4,     2;
  1,  12,    18,     6;
  1,  32,   108,    96,     24;
  1,  80,   540,   960,    600,   120;
  1, 192,  2430,  7680,   9000,  4320,   720;
  1, 448, 10206, 53760, 105000, 90720, 35280, 5040;

G. C. Greubel, Antidiagonals n = 0..50, flattened
F. A. Haight, Overflow at a traffic light, Biometrika, 46 (1959), 420-424. See page 422.
F. A. Haight, Overflow at a traffic light, Biometrika, 46 (1959), 420-424. (Annotated scanned copy)
F. A. Haight, Letter to N. J. A. Sloane, n.d.

Cf. A000012, A000142, A001563, A001787, A006043, A006044, A009998.
Cf. A055533, A072597 (antidiagonal sums), A089148, A119502, A152650.
Cf. A152656, A154372.

A152818:= func< n,k | (k+1)^(n-k)*Factorial(k)*Binomial(n,k) >;
[A152818(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 10 2023

len= 45; m= 1 + Ceiling[Sqrt[len]]; Sort[Flatten[#, 1] &[MapIndexed[ {(2 +#2[[1]]^2 +(#2[[2]] -1)*#2[[2]] +#2[[1]]*(2*#2[[2]] -3))/ 2, #1}&, Table[(k+1)^n*(n+k)!/n!, {n,0,m}, {k,0,m}], {2}]]][[All, 2]][[1 ;; len]] (* From Jean-François Alcover, May 27 2011 *)
T[n_, k_]:= (k+1)^(n-k)*k!*Binomial[n, k];
Table[T[n,k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 10 2023 *)

A(n,k) = (k+1)^n*(n+k)!/n! \\ Charles R Greathouse IV, Sep 10 2016

def A152818_row(n):
    R. = ZZ[]
    P = add((n-k+1)^k*x^(n-k+1)*factorial(n)/factorial(k) for k in (0..n))
    return P.coefficients()
for n in (0..12): print(A152818_row(n))  # Peter Luschny, May 03 2013

E.g.f. for array as a triangle: exp(x)/(1-t*x*exp(x)) = 1+(1+t)*x+(1+4*t+2*t^2)*x^2/2! + (1+12*t+18*t^2+6*t^3)*x^3/3! + .... E.g.f. is int {z = 0..inf} exp(-z)*F(x,t*z), (x and t chosen sufficiently small for the integral to converge), where F(x,t) = exp(x*(1+t*exp(x))) is the e.g.f. for A154372. - Peter Bala, Oct 09 2011
From Peter Bala, Oct 09 2011: (Start)
From the e.g.f., the row polynomials R(n,t) satisfy the recursion R(n,t) = 1 + t*sum {k = 0..n-1} n!/(k!*(n-k-1)!)*R(n-k-1,t). The polynomials 1/n!*R(n,x) are the polynomials P(n,x) of A152650.
Sum_{k=0..n} T(n, k) = A072597(n) (antidiagonal sums). (End)
From G. C. Greubel, Apr 10 2023: (Start)
T(n, k) = (k+1)^(n-k) * k! * binomial(n, k) (antidiagonal triangle).
Sum_{k=0..n} (-1)^k*T(n, k) = A089148(n). (End)

Better definition, extended and edited by Omar E. Pol and N. J. A. Sloane, Jan 05 2009

cp's OEIS Frontend

A080108 a(n) = Sum_{k=1..n} k^(n-k)*binomial(n-1,k-1).

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A072597 Expansion of 1/(exp(-x) - x) as exponential generating function.

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A152818 Array read by antidiagonals: A(n,k) = (k+1)^n*(n+k)!/n!.

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